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info.json
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{
"abstract": "Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $\\delta$. If $\\delta$ is too large, however, then counterexamples containing spurious local minima are known to exist. In this paper, we introduce a proof technique that is capable of establishing sharp thresholds on $\\delta$ to guarantee the inexistence of spurious local minima. Using the technique, we prove that in the case of a rank-1 ground truth, an RIP constant of $\\delta<1/2$ is both necessary and sufficient for exact recovery from any arbitrary initial point (such as a random point). We also prove a local recovery result: given an initial point $x_{0}$ satisfying $f(x_{0})\\le(1-\\delta)^{2}f(0)$, any descent algorithm that converges to second-order optimality guarantees exact recovery.",
"authors": [
"Richard Y. Zhang",
"Somayeh Sojoudi",
"Javad Lavaei"
],
"emails": [
"ryz@illinois.edu",
"sojoudi@berkeley.edu",
"lavaei@berkeley.edu"
],
"id": "19-020",
"issue": 114,
"pages": [
1,
34
],
"title": "Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery",
"volume": 20,
"year": 2019
}